from maxwell s equations to the theory of current source density analysis

Accepted Article DR COSTAS ANASTASSIOU (Orcid ID : 0000-0002-6793-0611) Received Date : 22-Mar-2016 Revised Date : 17-Jan-2017 Accepted Date : 30-Jan-2017 Article type : Special Issue Article From Maxwell’s equations to the theory of current-source density analysis Authors: Sergey L Gratiy1, Geir Halnes2, Daniel Denman1, Michael J Hawrylycz1, Christof Koch1, Gaute T Einevoll2,4, Costas A Anastassiou1,3 Allen Institute for Brain Science, Seattle, WA 98109, USA Faculty of Science and Technology, Norwegian University of Life Sciences, Aas, Norway Department of Neurology, University of British Columbia, Vancouver V5Z 3N9, British Columbia, Canada Department of Physics, University of Oslo, Oslo, Norway Keywords: electrical conductivity, extracellular recordings, field potentials, current transfer, electrical stimulation This article has been accepted for publication and undergone full peer review but has not been through the copyediting, typesetting, pagination and proofreading process, which may lead to differences between this version and the Version of Record Please cite this article as doi: 10.1111/ejn.13534 This article is protected by copyright All rights reserved Running title: The theory of current-source density analysis Accepted Article Abstract Despite the widespread use of current-source density (CSD) analysis of extracellular potential recordings in the brain, the physical mechanisms responsible for the generation of the signal are still debated While the extracellular potential is thought to be exclusively generated by the transmembrane currents, recent studies suggest that extracellular diffusive, advective and displacement currents—traditionally neglected—may also contribute considerably towards extracellular potential recordings Here, we first justify the application of the electro-quasistatic approximation of Maxwell’s equations to describe the electromagnetic field of physiological origin Subsequently, we perform spatial averaging of currents in neural tissue to arrive at the notion of the CSD and derive an equation relating it to the extracellular potential We show that, in general, the extracellular potential is determined by the CSD of membrane currents as well as the gradients of the putative extracellular diffusion current The diffusion current can contribute significantly to the extracellular potential at frequencies less than a few Hertz; in which case it must be subtracted to obtain correct CSD estimates We also show that the advective and displacement currents in the extracellular space are negligible for physiological frequencies while, within cellular membrane, displacement current contributes toward the CSD as a capacitive current Taken together, these findings elucidate the relationship between electric currents and the extracellular potential in brain tissue and form the necessary foundation for the analysis of extracellular recordings Introduction Electrical activity of excitable brain cells is realized by the transmembrane ionic currents This article is protected by copyright All rights reserved which, in turn, give rise to currents and the corresponding scalar electric potential in the Accepted Article extracellular space Measurements of extracellular potential therefore provide information about electrical activity in the brain and aid to unravel the function of the underlying neuronal circuits The high-frequency component (above ~500 Hz) of the extracellular potential, termed multi-unit activity, is typically used to detect spiking of individual neurons (Schmidt, 1984) In contrast, the signal at lower frequencies (below ~200 Hz), termed the local field potential (LFP) (Buzsáki et al., 2012; Einevoll et al., 2013), characterizes the collective electrical activity of neuronal populations At a spatial scale greater than that of a single cell, this collective electrical activity may be described by a spatially smooth threedimensional current-source distribution, termed current-source density (CSD) (Mitzdorf, 1985) The idea that the CSD may be estimated from the Laplacian of the extracellular potential recorded at nearby locations within the brain, originates from Pitts (1952) and forms the basis of CSD analysis Nicholson (1973) provided a theoretical justification for Pitt’s insight in the special case of the quasi-stationary approximation of Maxwell’s equations, which neglects both magnetic induction and displacement currents (Haus & Melcher, 1989) They tacitly assumed that tissue conductivity is independent of the frequency of the signal in the physiological range, and that diffusion, advection and displacement currents in the extracellular space are negligible in comparison to Ohmic drift current The validity of these assumptions, however, has been questioned in recent studies In an effort to explain the power spectrum of the LFP, Bédard & Destexhe (2009) developed a theoretical model predicting that ionic diffusion in the extracellular space is the main cause This article is protected by copyright All rights reserved for the frequency-dependence of the signal In a follow-up study, it was concluded that the Accepted Article CSD must be due to the extracellular diffusion current rather than the transmembrane currents (Bédard & Destexhe, 2011) Furthermore, analyzing the extracellular potential recordings, Riera et al (2012) found that the estimated laminar CSD profiles not sum to zero across the cortical depth as would be expected from their neuronal origin To address this paradox, they speculated that tissue polarization as well as diffusive and advective currents might need to be accounted for in the CSD analysis of extracellular potential recordings The limiting assumptions of the theory of CSD analysis (Nicholson, 1973; Nicholson & Freeman, 1975) and challenges to its validity from both experimentalists and theoreticians motivated us to revisit the physical basis of CSD analysis and examine its underlying assumptions Starting with Maxwell’s equations of macroscopic electromagnetism, we utilize the electro-quasistatic approximation to establish the equations describing fields of physiological origin We present the general relationship between currents and the potential in the extracellular space and motivate a coarse-grained description needed for the analysis of electrophysiological recordings Applying spatial averaging to currents in brain tissue, we arrive at the notion of the CSD of transmembrane currents and subsequently derive the equation for CSD analysis considering the possible frequency dependence of tissue conductivity We show that, in general, the extracellular potential is determined by the transmembrane currents as well as by the gradients of the putative extracellular diffusive currents, which can play an important role at the lowest frequencies In turn, the effect of the displacement and advective currents in the extracellular space is negligible as a result of fast charge relaxation However, within cells the displacement This article is protected by copyright All rights reserved Accepted Article current contributes toward the CSD as a capacitive current Materials and methods Electrophysiological recordings All surgeries and procedures were approved by the Allen Institute for Brain Science Institutional Animal Care and Use Committee Recordings were made in C57BL/6 male mice, > 12 weeks old (Jackson Laboratories, n = 2) Detailed descriptions of the experimental apparatus and procedures are available in a previously published report (Denman et al., 2016) Briefly, an initial surgery was made to attach a headpost to the skull Following surgery, the animal was allowed to recover for at least days before habituation Prior to recording, animals were allowed to fully habituate to head-fixation in the experimental apparatus over several sessions of increasing duration The apparatus consisted of a horizontal disc suspended in a spherical environment onto which light was projected Animals were allowed to run freely on the disc while head-fixed On the day of recording anesthesia was induced and maintained with inhaled isoflurane (5% induction, 2-3% maintenance) A small craniotomy was made over primary visual cortex using stereotactic coordinates and a reference screw was implanted as far from the recording site as possible, rostrally, within the area of exposed skull The animal was transferred to the experimental apparatus and allowed to recover from anesthesia A highdensity array of extracellular electrodes, containing electrodes spaced every 20 µm vertically (Lopez et al., 2016), was lowered through the craniotomy; the dura matter was pierced by the electrode array The array insertion continued until some electrodes were below the cortex and within underlying structures At this level, several electrodes remained This article is protected by copyright All rights reserved above the pial surface, ensuring complete coverage of cortex After reaching this insertion Accepted Article depth, the electrode was allowed to rest untouched for at least 30 minutes before data were recorded Visually activity was evoked in cortex using brief full-field luminance changes Luminance changes were 50 milliseconds in duration and alternated between increases and decreases in luminance, returning to a mean luminance (~3 cd/m2) for seconds between changes The magnitude of luminance changes was 0.2 cd/m2 for OFF and 5.8 cd/m2 for ON Signals were acquired in two parallel data streams at 10-bit resolution: a MUA data stream highpass filtered at 500 Hz and sampled at 30 kHz and a LFP data stream low-pass filtered at 300 Hz and sampled at 2.5 kHz The analyses presented were performed on the LFP data stream Estimation of the CSD The array data was mapped to the cortical depth locations after identifying the channel corresponding to the pial surface by visual inspection of raw LFPs post-hoc Brief (~500 msec) chunks of raw data from each channel were plotted in an arrangement that allowed comparison of neighboring channels; the channel at which amplitude dropped discontinuously and higher frequency components became more homogenous was chosen as the pial surface The CSD was estimated from the trial-averaged cortical LFP recordings for both ON (n=50) and OFF (n=50) luminance conditions To estimate the CSD we used a variant of the deltasource iCSD method (Pettersen et al., 2006) assuming a radius of 0.5 mm for the circularlysymmetric sources around the recording electrode This method utilizes the solution of the Poisson equation, Eq (22), for the extracellular potential Φ at the i-th cortical location It can be expressed as a linear superposition Φ = of sources This article is protected by copyright All rights reserved at each of j-th location, is a forward operator Correspondingly, the CSD may be estimated as ̂ = where is the regularized inverse of the forward operator, which suppresses the Accepted Article where Φ, contribution of the noise on the estimated sources (Gratiy et al., 2011) The tissue conductivity was taken at 0.3 mS/mm (Wagner et al., 2014) The divergence of the diffusive current in Eq (21) when expressed in terms of the ionic concentrations, is given by − ∙ 〈 〉 = ∑ ∙( 〈 〉 ), where 〈 〉 is the coarse- grained concentration of the i-th ionic species Assuming K+ and Na+ ions dominate the changes in the ionic concentration and utilizing the condition of electroneutrality (Δ[K +] + Δ[Na +] = 0), we find − ∙ 〈 〉 = ( − ) [K+] This constitutes a Poisson equation for [K+] Therefore, the divergence of the diffusive current (i.e., the apparent CSD resulting from diffusion) may be estimated from measurement of [K+] , applying the same technique as for estimating the CSD from the LFP recordings Similarly, we assume that the diffusion current is localized to the same cylindrical volume as the CSD and varies only along the cortical depth We use = 1.33 ∙ 10 = 1.96 ∙ 10 m2/s and m2/s (Grodzinsky, 2011) Results Equations of electromagnetisms of physiological origin Our starting point is the set of macroscopic Maxwell’s equations describing electromagnetic field variables, which are spatially averaged over volumes that are large compared to atomic volumes (Russakoff, 1970; Griffiths, 2012): This article is protected by copyright All rights reserved Accepted Article × where =− × = + , (1) , (2) ∙ = , (3) ∙ = 0, (4) and are the free (i.e., unbound) charge density and current density, and are the electric and magnetic fields, respectively The effects of bound charges and currents are included in the auxiliary fundamental and and fields, which may be expressed in terms of the fields using constitutive relations For linear materials with instantaneous response properties, it holds that permittivity and = and = where is the electric the magnetic permeability of the medium Spatial averaging over volumes including many atoms eliminates references to individual atoms and removes the high spatial frequency components of the field variables Correspondingly, the macroscopic description may be viewed as a description for which the spatial Fourier component of the field variables above some limiting frequency are irrelevant and eliminated by performing averaging over volumes with the dimension ~1/ The irrelevant spatial frequencies are determined not by the physical structure of the system, but rather by the particular problem we are attempting to solve (Robinson, 1971) As such, the macroscopic equations for a particular system may be formulated using different averaging volumes –all depending on the spatial scales relevant for the application to a particular problem Maxwell’s equations describe a host of electromagnetic phenomena occurring across a wide range of spatial and temporal scales and are difficult to analyze in a general form To This article is protected by copyright All rights reserved describe the electric fields in the brain, we introduce two approximations which drastically Accepted Article simplify the mathematical treatment of electrodynamics Firstly, for fields of physiological origin, the typical temporal frequencies are so low (less than a few thousand Hz) that the magnetic induction has a negligible effect on the electric field (Plonsey & Heppner, 1967; Rosenfalck, 1969) The error in the electric field at angular frequency induction is given by relative to the actual field / ~( made by neglecting the magnetic ) (Haus & Melcher, 1989) Here = / is the time it takes the electromagnetic wave to propagate across the characteristic length at velocity = /√ in a material having relative permittivity and permeability , where is the speed of light in vacuum For example, in grey matter we may take the characteristic length ~1 mm, corresponding to the cortical thickness Using measured values of permittivity and permeability in mammalian grey matter (e.g., Wagner et al., 2014), yields the relative error / < 10 for frequencies in a range of 10 Hz to 10 kHz, so that magnetic induction can be safely neglected Neglecting the magnetic induction in Faraday’s law constitutes the electro-quasistatic approximation (Haus & Melcher, 1989): × ≈0⇒ =− , (5) i.e., the electric field is essentially conservative and can be expressed as a gradient of a scalar potential Consequently, using the electro-quasistatic approximation to describe fields in brain tissue of physiological origin amounts to a negligible error when compared to the exact solution using a full set of Maxwell’s equations In contrast, the displacement current in Ampere-Maxwell’s law (Eq 2) is responsible for the capacitive charging of neural membranes and cannot be neglected This article is protected by copyright All rights reserved Secondly, the macroscopic velocity of ions in the brain and the magnetic field of Accepted Article physiological origin are so low that the magnetic component of the Lorentz force = ( + × ) is negligible Indeed, using the largest bulk flow velocity ~1 m/sec due to the arterial blood flow (Bishop et al., 1986), the typical magnetic field ~100 fT (Hämäläinen et al., 1993) and extracellular electric field ~1 V/m (Cordingley & Somjen, 1978) arising from neuronal activity, yields / ~10 , i.e., the force due to the magnetic field is negligible Consequently, the effect of the magnetic field of physiological origin on the motion of free charges and the corresponding current density may be neglected The negligibility of the the magnetic induction and magnetic component of the Lorentz force results in the decoupling of the electric and magnetic fields Since the current density is now independent of the magnetic field, it is convenient to eliminate the field from consideration by taking the divergence of Eq (2), resulting in a current continuity statement: where the total current density ∙ ≝ + = 0, (6) is solenoidal, i.e., current travels along closed loops Current continuity, Eq (6), also represents the principle of charge conservation, which may be cast in a familiar form ∙ + displacement current in terms of the density of free charges = by expressing the using Gauss’s law, Eq (3) Together with the constitutive relations, Eqs (3), (5) and (6) determine the electric field, current density and charge density Then, if desired, the magnetic field can be determined from the known current density by using Eqs (2) and (4) This article is protected by copyright All rights reserved experimental recordings Accounting for all possible mechanisms of charge transfer we Accepted Article show that, in general, both the CSD as well as gradients of extracellular diffusion currents determine the coarse-grained extracellular potential, which satisfies the Poisson equation The advective and displacement current in the extracellular place may be neglected for frequencies of physiological origin On the other hand, the displacement current in the cellular space is included as a part of CSD The practice of CSD analysis Depending on the assumptions about properties and processes occurring in brain tissue, either Eq (20), (21) or (22) could be used to estimate the CSD of membrane currents from the recorded extracellular potential When neither the frequency dependence of conductivity nor the diffusion currents can be neglected, Eq (20) must be used to estimate the CSD of membrane currents separately at each temporal frequency Only if the frequency dependence of extracellular conductivity is negligible can Eq (21) be used In the simplest situation of constant conductivity and no diffusion currents, Eq (22) may be used instead Conversely, when diffusion currents cannot be neglected, one must estimate them independently and, per Eq (21), subtract them from the Laplacian to arrive at the CSD of membrane currents When applied to experimental data, one must overcome a number of issues (Freeman & Nicholson, 1975): 1) electrical recordings are rarely available to evaluate the Laplacian across all three spatial dimensions, necessitating additional assumptions regarding the source distribution; 2) noise in the experimental data is significantly amplified by spatial differentiation, requiring the use of noise regularization strategies These challenges This article is protected by copyright All rights reserved motivated the development of several methods and respective software tools for estimating Accepted Article the CSD (Pettersen et al., 2006; Łęski et al., 2007, 2011; Potworowski et al., 2012) which have been successfully applied to analyze experimental recordings Both the coarse-grained extracellular potential and the CSD in the governing equation utilize the same averaging kernel and correspondingly have the same spatial resolution Therefore, the spatial resolution of the estimated CSD is determined by the spatial resolution of the data, i.e., the inter-channel spacing on the electrode shank Whether the selection of a particular spacing is sufficient for a particular application depends on the spatial frequencies of interest, which need to be resolved The spatial localization of the underlying neuronal currents is in part determined by the temporal dynamics of the electrical activity because of the frequency dependence of the membrane impedance For instance, the membrane length constant in passive dendrites is inversely proportional to the temporal frequency (Koch, 2004) such that membrane currents at higher temporal frequencies decay faster along the cable (i.e., more localized) than those at lower temporal frequencies (Lindén et al., 2010; Anastassiou et al., 2015) Correspondingly, as manufacturing technology continues to advance towards increasing channel density, more localized current sources can be resolved, which are typically characterized by faster temporal dynamics Difference between the fine-grained and coarse-grained descriptions of extracellular potential The limited spatial resolution of extracellular recordings dictates the need for a coarsegrained description of membrane current sources and the extracellular potential in brain tissue Here we developed such a description by performing spatial averaging of currents This article is protected by copyright All rights reserved while distinguishing between the cellular and extracellular spaces Applying the Accepted Article mathematical identity developed in Appendix A, the coarse-grained currents within the cellular space were expressed via a volume density of transmembrane currents, i.e., the CSD Consequently, we find that generally the Poisson equation describing the coarsegrained extracellular potential, Eq (21), includes both the CSD of transmembrane currents and the divergence of the diffusion currents as sources on the right-hand side In contrast, Eq (10) describes the fine-grained extracellular potential and may only include the divergence of diffusion currents as a source on the right-hand side The difference between the two descriptions lies in the way they account for the boundary conditions, i.e., the transmembrane currents The solution of Eq (10) for the extracellular potential is sought within the narrow confines of extracellular space and the effects of the transmembrane currents are included through the boundary conditions On the other hand, the solution of Eq (21) is sought within the tissue and the contribution of the membrane currents is included in the CSD Furthermore, Eq (10) includes the conductivity of extracellular space while Eq (21) includes tissue conductivity Diffusion currents in the extracellular space Several pathological conditions, such as hypoxia, anoxia, ischemia and spreading depression are associated with significant ion concentration changes in the extracellular space (Syková & Nicholson, 2008) Also during non-pathological conditions, neural signaling may cause local ion-concentration changes For example, [K+]o elevations in the cat striate cortex in response to bright bars moving across the receptive field amount to ~0.1 mM (Connors et al., 1979), whereas strong repeated cortical stimulation may locally elevate [K+]o up to 10 mM (Pumain & Heinemann, 1985) This article is protected by copyright All rights reserved Extracellular ion concentration changes are typically inhomogeneous across the cortical Accepted Article depth (Cordingley & Somjen, 1978; Nicholson et al., 1978; Pumain & Heinemann, 1985) The presence of ionic concentration gradients results in ionic diffusion, which in turn gives rise to electrical current in the extracellular space As for temporal dynamics, extracellular [K+] builds-up and clears with a time constant ≳1 s (Cordingley & Somjen, 1978; Connors et al., 1979) Diffusion currents due to extracellular concentration gradients are thus likely to change at a slow time scale of seconds and correspondingly are expected to contribute only to the low frequency components of the LFP In the present application, we found that extracellular diffusion may be of importance for determining the LFP at frequencies ≲ Hz (see Sec Effects of extracellular diffusion on the LFP recordings) for physiological conditions accompanied by strong (≳1 mM) changes in the extracellular ionic concentrations Similar results were found in a previous computational study, where diffusive currents were found to influence LFP frequency components up to a few Hz in the case of large extracellular concentration gradients (Halnes et al., 2016) How these findings change our interpretation of the depth LFP with regards to ongoing activity? Being a low-frequency effect, extracellular diffusion is unlikely to play a role in oscillations such as theta (2-12 Hz), beta (12-30 Hz), gamma (30-80 Hz), etc On the other hand, several slower oscillatory patterns exist with their main frequency component being below Hz such as slow neocortical rhythms and delta waves (Gloor et al., 1977; Buzsáki et al., 1988; Steriade, McCormick, et al., 1993) These patterns have been shown to play key role in neural functioning and coordination For example, slow neocortical activity with its accompanying UP-DOWN states critically contributes to the temporal organization of other cortical patterns, such as sleep spindles, gamma oscillations and K-complexes (Achermann & This article is protected by copyright All rights reserved Borbely, 1997; Steriade & Amzica, 1998; Mölle et al., 2002; Mukovski et al., 2007) as well as Accepted Article hippocampal sharp wave ripples (Sirota et al., 2003; Sirota & Buzsáki, 2005) Hitherto, the source of the extracellular signal associated with slow neocortical oscillations has been chiefly ascribed to intracellular UP-DOWN dynamics and rhythmic polarization (extending 10-20 mV) of cortical neurons Yet, a progressive decrease in extracellular calcium concentration by approximately 20% has also been measured during UP states It has been hypothesized that such Ca-concentration changes can lead to decrease in neurotransmitter release probability and, eventually, promote the subsequent DOWN state (Massimini & Amzica, 2001) Yet, our work suggests an additional role of such Ca-concentration change as they can affect the CSD estimates Our study suggests an alternative interpretation of the signals associated with slow neocortical activity where significant part of the signal below Hz may be contributed by ionic diffusion with the rest of it associated with neural membrane polarization Here we propose a way to account for the effects of extracellular ionic diffusion on the extracellular potential The theory shows that the extracellular potential at the coarsegrained scale is governed by Poisson’s equation (Eq 21) with the source term generally including the membrane currents and contributions from extracellular diffusion When the diffusion current gradients can be neglected, the extracellular potential is still determined by Poisson’s equation in accordance with the original theory of CSD analysis (Nicholson, 1973) As such, our theory is the generalization of the original theory of CSD analysis Nevertheless, our findings contrast with the theory of Bédard & Destexhe (2011), who predicted that the coarse-grained extracellular potential is governed by Poisson’s equation only when the diffusion effects are included (see Eq (11) in Bédard & Destexhe (2011)), This article is protected by copyright All rights reserved otherwise the extracellular potential is governed by the Laplace equation Consequently, it Accepted Article would follow that the CSD estimated by computing a Laplacian of the extracellularly recorded potential must be interpreted as the divergence of the extracellular diffusion currents Naturally, such finding would suggest an essential role for extracellular ionic diffusion currents in determining the extracellular potential However, we believe that this conclusion is erroneous Equation (11) in Bédard & Destexhe (2011) has the same physical meaning as Eq (10) in this paper As discussed in the Sec “Fine-grained description of electric currents in the extracellular space”, the application of Eq (11) to describe the extracellular potential requires solving it within the narrow confines of the extracellular space and consequently requires specifying the boundary conditions along the cellular membrane The need for the explicit boundary condition along cellular membrane makes Eq (10) and correspondingly Eq (11) in Bédard & Destexhe (2011) unsuitable for the analysis of extracellular recordings at the coarse-grained scale In contrast, here we developed the formalism for describing the extracellular potential at the coarse-grained scale within a tissue space by incorporating the membrane currents into the CSD term in the Poisson equation that makes it suitable for the analysis of LFP recordings Appendix A: Coarse-graining Here we develop the expression for the divergence of a vector field averaged over a volume bound by a generally convoluted and tortuous surface (or a collection of surfaces) such as neuronal membranes Subsequently, we will use the derived expression to formally define the transmembrane CSD Let’s consider an arbitrary differentiable vector field ( ) and define the averaged field over the volume : This article is protected by copyright All rights reserved Accepted Article 〈 ( )〉 ≝ ′ ( ′) ( − bounded by an arbitrarily torturous surface (23) ) such as cellular membrane To simplify the notation, we will use Einstein’s convention for index summation such that the divergence ∙ ≡∇ position , where ∇ = is the derivative with respect to the component of the = ( , , ) Taking the divergence of the vector field in Eq (23) we find: ∇ 〈 ( )〉 = ′ ( ′)∇ ( − ), (24) where the derivative may be moved inside the integration since differentiation and averaging act on different variables Noting that ∇ ∇ 〈 ( )〉 = − ′ ( )∇′ ( − ) = −∇′ ( − ), ( − ) we have (25) where now the differentiation and integration are performed with respect to the primed coordinates Since the integrand on the right-hand side may be expressed as: − ( )∇ ( − ) = −∇ ( − ) ( ) + ( − )∇ ( ), (26) we find that ∇ 〈 ( )〉 = − ′∇′ ( − ) ( ) + ′ ( − )∇′ ( ) (27) Applying Gauss’ theorem to the first term and the definition of the averaged field, Eq (22), to the second term on the right-hand side, we arrive at the identity expressed in vector notation: where we put ∙ 〈 ( )〉 = − ≝ ’ with ∙ ( ) ( − ) + 〈 ∙ ( )〉 , (28) being a unit surface normal The appearance of the This article is protected by copyright All rights reserved surface term on the right-hand side indicates that, in general, the averaging and Accepted Article differentiation not commute! Commonly, averaging is performed over the entire space such that the bounding surface is infinitely removed from the region of interest in space Since the averaging kernel is nonzero only within a neighborhood | − |< around each field point , the surface term vanishes, resulting in commutativity between averaging and differentiation: ∙ 〈 ( )〉 = 〈 ∙ ( )〉 (29) Conversely, when the bounding surface crosses the neighborhood | − | < , we must use a general identity, Eq (28) In particular, this applies when averaging fields over the cellular space, because there the averaging volume includes many bounding surface (i.e, surface membranes) as illustrated in Fig B The surface term in Eq (28) is essential when applied to the total current density ( ) Since the total current density is solenoidal, ∙ = 0, we find: ∙ 〈 ( )〉 = − ′ ( ) ( − where we defined the membrane current density current density averaged over a volume by the averaging kernel—over the surface ≝ ), (30) ∙ Equation (29) states that equals a negative sum of the currents—weighted bounding the volume If represents a cellular membrane then, Eq (30) states that the current density averaged over the cellular volume ∙ 〈 ( )〉 equals the negative weighted sum of the transmembrane currents density ( ) Appendix B: On the use of the electroneutrality assumption in the This article is protected by copyright All rights reserved presence of diffusion fluxes Accepted Article In the Sec “Fine-grained description of electric currents in the extracellular space” we used the current continuity Eq (6) to derive Eq (10) describing the extracellular potential in the presence of diffusion On the other hand, applying the EQS approximation to the Gauss’ law we find: ∙ (ε )=− , (31) which must also be satisfied by the extracellular potential For physiological frequencies ≪ and from Eq (9) we find the charge density: where = = + ∙( ), (32) includes the contributions of the fixed charges matrix and mobile charges = ∑ on the extracellular in the interstitial fluid Assuming, for simplicity, the homogeneous electrical properties and substituting the expression for the charge density in Eq (32) into Eq (31) we find: σ∇ =− ∙( ), (33) which is special case of Eq (10) with constant conductivity Thus, both Gauss’s law and the charge continuity result in consistent equations for describing the extracellular potential On the other hand, if exact electroneutrality were to be assumed, i.e (31) we would end up with the Laplace equation ∇ = 0, then from Eq = 0, leading to an inconsistency with Eq (33) and, correspondingly, a more general Eq (10) Thus, strict electroneutrality cannot be assumed for the purposes of describing the electric field in the presence of diffusion fluxes This article is protected by copyright All rights reserved Acknowledgments Accepted Article This work was funded by the Allen Institute for Brain Science, the Swiss National Science Foundation, the National Institute of Neurological Disorders and Stroke, the Mathers Foundation and the Research Council of Norway S L Gratiy, D Denman, C Koch, M J Hawrylycz and C A Anastassiou thank the Allen Institute founders, Paul G Allen and Jody Allen, for support The authors declare no conflicts of interest, financial or otherwise Abbreviations LFP – local field potential, CSD –current source density, EQS—electro-quasistatic Figure legends Figure Spatial averaging of currents in brain tissue (A) An example of a kernel ( ), which may be used in the spatial averaging procedure The width of the plateau is much larger than the size of the dendritic diameter The function transitions to zero monotonically in a well-behaved fashion to avoid jitter in the averaged variables (B) Schematic of a crosssection of neural tissue with neuropil (grey) surrounded by the extracellular space (white) Spherical averaging volume (shown by a dashed black circle) with an effective radius corresponding to the width of the averaging kernel encloses multiple processes of several nearby cells The CSD at the central location (black dot) is computed by summing membrane currents over the spherical averaging volume The CSD computed over this volume may generally result from a combination of outward (red outline) and inward (blue outline) membrane currents (C) Schematic of cellular current from a dotted rectangular detail in (B) The axial currents along the neuropil are shown as crosses or dots corresponding to the flow into or out of the paper, respectively Some of the cytoplasmic current diverts towards This article is protected by copyright All rights reserved the membrane (black arrows) and results in the outward (red arrows) or inward (blue Accepted Article arrows) transmembrane current According to Eq (15), the divergence of the averaged currents in the cellular space (black lines) may be found as a as weighted sum of the transmembrane currents Figure Estimation of the contribution of extracellular diffusion towards the LFP recordings (A) Trial-averaged LFP recordings (left) and the corresponding CSD estimates (right) from the mouse visual cortex in response to the presentation of the full-field 50 ms flash: “ON flash” (top) and “OFF flash” (bottom) Black vertical line indicates the stimulus onset (B) The experimentally recorded spatial profile of the extracellular [K+] in cat visual cortex in response to the electrical stimulation of the thalamus (Reproduced by permission from Cordingley & Somjen (1978), Fig 5) and the corresponding assumed [Na+] profile (top left) The estimated spatial profile of the apparent CSD resulting from diffusion (top right) The modeled [K+] transients corresponding to the experimentally recorded half-decay times (Cordingley & Somjen, 1978) for the post-stimulus clearance (bottom) (C) Comparison of the power spectral density of the estimated apparent CSD resulting from diffusion (B) for the thee modeled half-decay times (red, green and blue lines) and of the CSD from (A) averaged over the cortical depths (mean: 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Rushmore, J., Russo, C.J., Dipietro, L., Fregni, F., Simon, S., Rotman, S., Pitskel, N.B., Ramos-Estebanez, C., Pascual-Leone, A., Grodzinsky, A.J., Zahn, M., & Valero-Cabré, A (2014) Impact of brain tissue filtering on neurostimulation fields: A modeling study NeuroImage, Neuro-enhancement, 85, Part 3, 1048–1057 This article is protected by copyright All rights reserved Accepted Article This article is protected by copyright All rights reserved ... title: The theory of current- source density analysis Accepted Article Abstract Despite the widespread use of current- source density (CSD) analysis of extracellular potential recordings in the brain,... by Poisson? ?s equation in accordance with the original theory of CSD analysis (Nicholson, 1973) As such, our theory is the generalization of the original theory of CSD analysis Nevertheless, our... motivated by the limiting and implicit assumption of the theory of CSD analysis and the ensuing challenges to its validity The original theory of CSD analysis (Nicholson, 1973) was developed in the limit